Transitive Closure via Composition of Relations

Abstract

This work explored the fundamental aspects of binary relations, specifically transitive closure and how it was computed through the composition of relations. Our starting point was to construct a strong foundation in the definitions, properties, and representations of relations, both visually and numerically. The composition of relations was analyzed in detail to demonstrate its role in constructing transitive closures. The matrix-based approach was emphasized, leading to a formal presentation of Warshall’s algorithm. Finally, a comparative analysis of computation methods was provided to highlight the strengths and limitations of each approach. The study strengthened key theoretical concepts and presented a structured method for understanding transitive closure in the context of discrete mathematics and theoretical computer science. Extending these methods to fuzzy relations and analyzing the optimization of transitive closure computation in large-scale data structures and dynamic systems is a promising direction for future research.